1. Representation of real numbers

2. What are Real Numbers? π Real Numbers include:
Whole Numbers (like 1,2,3,4, etc)
Rational Numbers (like 3/4, 0.125, , 1.1, etc )
Irrational Numbers (like π, √3, etc )
Real Numbers can also be positive, negative or zero
In Computing, real numbers are also known as floating point numbers
14.75
3148
22.9
99/100 15 π
2/7
-27
4/5 √3

3. Standard Form
Standard form is a scientific notation of representing numbers as a base number and an exponent.
Using this notation:
The decimal number can be represented as
x 103, with mantissa = , base = 10 and exponent = 3
The decimal number can be represented as
x 102, with mantissa = , base = 10 and exponent = 2
The decimal number can be represented as
6.34 x 10-3, with mantissa = 6.34, base = 10 and exponent = -3
Any number can be represented in any number base in the form m x be

4. Floating Point Notation
In floating point notation, the real number is stored as 2 separate bits of data
A storage location called the mantissa holds the complete number without the point.
A storage location called the exponent holds the number of places that the point must be moved in the original number to place it at the left hand side.

5. Floating Point Notation
What is the exponent of ?
The exponent is 5, because the decimal point has to be moved 5 places to get it to the left hand side.
The exponent would be represented as 0101 in binary

6. Floating Point Notation
How would be stored using 8 bits for the mantissa and 4 bits for the exponent?
We have already calculated that the exponent is 5 or 0101.
= x 25 = x 20101
It is not necessary to store the ‘x’ sign or the base because it is always 2.
Mantissa
Exponent

7. Floating Point Notation
How would 24.5 be stored using 8 bits for the mantissa and 4 bits for the exponent?
In binary, the numbers after the decimal point have the following place values:
1/2 1/4 1/8 1/ / / /128
24 has the binary value 11000
0.5 (or 1/2) has the binary value .1
24.5 =

8. Floating Point Notation
How would be stored using 8 bits for the mantissa and 4 bits for the exponent?

9. Floating Point Notation
How would be stored using 8 bits for the mantissa and 4 bits for the exponent?
The exponent is 7 because decimal point has to move 7 places to the left
= x 27 = x 20111
Mantissa
Exponent

10. Accuracy
Store in floating point representation, using 8 bits for the mantissa and 4 bits for the exponent.
Mantissa
Exponent
The mantissa only holds 8 bits and so cannot store the last two bits
These two bits cannot be stored in the system, and so they are forgotten.
The number stored in the system is which is less accurate that its initial value.

11. Accuracy
If the size of the mantissa is increased then the accuracy of the number held is increased.
Mantissa (10 bits)
Exponent

12. Range
If increasing the size of the mantissa increases the accuracy of the number held, what will be the effect of increasing the size of the exponent?
Using two bits for the exponent, the exponent can have the value 0-3
Mantissa
Exponent (2 bits)
This means the number stored can be in the range
(0) to
( )

13. Range
Increasing the exponent to three bits, it can now store the values 0-7
Mantissa
Exponent (3 bits)
This means the number stored can be in the range
(0) to
(127.5)
If the size of the exponent is increased then the range of the number s which can be stored is increased.